See "A cardinal preserving immune partition of the ordinals" by Mack Stanley, Fundamenta Mathematicae 148 (1995), 199-221

An infinite set (or class) of ordinals is said to be immune if it neither contains nor is disjoint from any infinite constructible subset of its supremum. The main result of the paper reads as follows: There exists an immune class of ordinals in a cardinal-preserving and GCH-preserving class generic extension of L.

Stanley, M. C., A cardinal preserving immune partition of the ordinals, Fundam. Math. 148, No. 3, 199-221 (1995). ZBL0843.03028.

An infinite set (or class) of ordinals is said to be immune if it neither contains nor is disjoint from any infinite constructible subset of its supremum. The main result of the paper reads as follows: There exists an immune class of ordinals in a cardinal-preserving and GCH-preserving class generic extension of L.